Lecture 1: Motivation, Intuition, and Examples
Outline: Motivation, definition, and intuition behind metric spaces. Redefining 18.100A Real Analysis and 18.100P Real Analysis in terms of metrics: open/closed sets, convergence, Cauchy sequences, and continuity.
Lecture 2: General Theory
Outline: Some general theory of metric spaces regarding convergence, open and closed sets, continuity, and their relationship to one another.
Lecture 3: Compact Sets in Rⁿ
Outline: Norms and analysis on finite sets (as motivation for compact sets). Topological compactness and sequential compactness. Today will focus purely on theorems regarding compact subsets of Rⁿ. The Heine-Borel theorem and the Bolzano-Weierstrass theorem.
Lecture 4: Compact Metric Spaces
Outline: Compact sets on general metric spaces. Showing sequential compactness is equivalent to topological compactness, which is equivalent to being totally bounded and complete (on metric spaces).
Lecture 5: Complete Metric Spaces
Outline: Completions of metric spaces, motivating Lp spaces, Sobolev spaces, p-adic numbers, Banach spaces, and Hilbert spaces.
Lecture 6: Where We Go from Here
Outline: How these topics were motivated, and a preview of how these topics come up in later classes.